A Borsuk Theorem for Antipodal Links and a Spectral Characterization of Linklessly Embeddable Graphs1 LÁszlÓ LovÁsz2 and Alexander Schrijver3

A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs1 László Lovász2 and Alexander Schrijver3 Abstract. For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that µ(G) ≤ 4 if and only if G is linklessly embeddable (in 3 R ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuk-type theorem on the existence of a pair of antipodal linked (k − 1)- 2k−1 spheres in certain mappings φ : S2k ! R . This result might be of interest in its own right. We also derive that λ(G) ≤ 4 for each linklessly embeddable graph G = (V; E), where λ(G) is the graph paramer introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension V of any subspace L of R such that for each nonzero x 2 L, the positive support of x induces a nonempty connected subgraph of G.) 1. Introduction Motivated by estimating the maximum multiplicity of the second eigenvalue of Schrödin- ger operators, Colin de Verdière [4] (cf. [5]) introduced an interesting new invariant µ(G) for graphs G, based on spectral properties of matrices associated with G. He showed that the invariant is monotone under taking minors (that is, if H is a minor of G then µ(H) ≤ µ(G)), that µ(G) ≤ 1 if and only if G is a disjoint union of paths, that µ(G) ≤ 2 if and only if G is outerplanar, and that µ(G) ≤ 3 if and only if G is planar. In this paper we show that µ(G) ≤ 4 if and only if G is linklessly embeddable in R3. An embedding of a graph G in R3 is called linkless if each pair of disjoint circuits in G are unlinked closed curves in R3 (for our ourposes, the following definition of linking suffices: two disjoint curves are unlinked, if there is a mapping of the unit disc into R3 such that its boundary is mapped onto the first curve and the image of the disc is disjoint from the second). G is linklessly embeddable if G has a linkless embedding in R3. (Èmbedding' presumes that vertices and edges have disjoint images.) Our result was conjectured by Robertson, Seymour, and Thomas [9], and has the following context. Robertson, Seymour, and Thomas [10] showed that a graph G is linklessly embeddable if and only if G does not have any of the seven graphs in the Petersen family as a minor | the Petersen family consists of all graphs obtainable from K6 by so-called ∆Y- and Y∆-operations (it includes the Petersen graph). (Y∆ means deleting a vertex of degree 3, and making its three neighbours mutually adjacent. ∆Y is the reverse operation (applied to a triangle).) 11991 Mathematics Subject Classification: primary: 05C10, 05C50, 57M15, secondary: 05C50, 57N15 2Department of Computer Science, Yale University, New Haven, Connecticut 06520, U.S.A. email: [email protected] 3CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands and Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. Research partially done while visiting the Department of Computer Science at Yale University. email: [email protected] 1 Since any of the graphs G in the Petersen family has µ(G) ≥ 5 (Bacher and Colin de Verdière [1]), it follows that if µ(G) ≤ 4 then G is linklessly embeddable. So we prove the reverse implication. Thus, next to the combinatorial characterization (in terms of minors) of the (topologically defined) class of linklessly embeddable graphs, there is a spectral characterization. Our proof method also applies to a related parameter called λ(G), introduced by van der Holst, Laurent, and Schrijver [7]. It is defined as follows. Let G = (V; E) be a graph. Call V a linear subspace L of R a representation of G if for each x 2 L, supp+(x) is nonempty and Gjsupp+(x) is a connected graph. Here we use the following notation. If U ⊆ V then GjU is the subgraph of G induced by U (and G − U = Gj(V n U)). For any vector x 2 RV , supp(x) is the support of x, that is, supp(x) := fv 2 V jx(v) 6= 0g. The positive support is supp+(x) := fv 2 V jx(v) > 0g and the negative support is supp−(x) := fv 2 V jx(v) < 0g. Now by definition, λ(G) is the maximum dimension of any representation L of G. It is easy to see that λ(G) is monotone under taking minors. Moreover, if G is a clique sum of graphs G1 and G2 (that is, if G arises from G1 and G2 by identifying a clique in G1 and G2), then λ(G) = maxfλ(G1); λ(G2)g. In [7] it is shown that λ(G) ≤ 1 if and only if G is a forest, that λ(G) ≤ 2 if and only if G is series-parallel, and that λ(G) ≤ 3 if and only if G arises by taking clique sums and subgraphs from planar graphs. In this paper we show that λ(G) ≤ 4 if G is linklessly embeddable | hence, more generally, if G arises by taking clique sums and subgraphs from linklessly embeddable graphs. We do not know if the reverse implication holds. A key ingredient in our proof is a Borsuk-type theorem on the existence of antipodal links, which we formulate and prove in Section 2. We derive it from an extension of a theorem of Bajmóczy and Bárány [2] establishing a polyhedral form of Borsuk's antipodal theorem. In Section 3 we derive that λ(G) ≤ 4 for linklessly embeddable graphs G. In Section 4 we give some preliminaries on the Colin de Verdère parameter µ(G), and after that, in Section 5, we show that µ(G) ≤ 4 for linklessly embeddable graphs G. Finally, in Section 6 we consider a number of open questions related to µ(G) and λ(G). 2. A Borsuk-type theorem for antipodal links Let P be a convex polytope in Rn. We say that two faces F and F 0 are antipodal if there exists a nonzero vector c in Rn such that the linear function cT x is maximized by every point of F and minimized by every point of F 0 (Figure 1). So F and F 0 are antipodal if and only if F − F 0 is contained in a face of P − P . Call a continuous map φ of a cell complex into Rm generic if the images of a k-face and an l-face intersect only if k + l ≥ m, and for k + l = m they have a finite number of intersection points, and at these points they intersect transversally. (In this paper, faces are relatively open.) For any convex polytope P in Rn, let @P denote its boundary. The following theorem extends a result of Bajmóczy and Bárány [2]. (The difference is that their theorem concludes that φ(F ) \ φ(F 0) is nonempty. Their proof uses Borsuk's theorem. We give an independent proof.) 2 Theorem 1. Let P be a full-dimensional convex polytope in Rn and let φ be a generic continuous map from @P to Rn−1. Then there exists a pair of antipodal faces F and F 0 with dim(F ) + dim(F 0) = n − 1 such that jφ(F ) \ φ(F 0)j is odd. Proof. We prove a more general fact. Call two faces parallel if their projective hulls have a nonempty intersection that is contained in the hyperplane at infinity. So faces F and F 0 are parallel if and only if their affine hulls are disjoint while F − F and F 0 − F 0 have a nonzero vector in common. (Note that two antipodal faces are parallel if dim(F ) + dim(F 0) ≥ n.) Figure 1: Parallel and non-parallel antipodal faces Now it suffices to show (1) Let P be a full-dimensional convex polytope in Rn having no parallel faces and let φ be a generic continuous map from @P to Rn−1. Then 0 X jφ(F ) \ φ(F )j is odd, where the summation extends over all antipodal pairs fF; F 0g of faces. (It would be enough to sum over all antipodal pairs fF; F 0g with dim(F )+dim(F 0) = n−1.) To see that it suffices to prove (1), it is enough to apply a random projective transformation close to the identity. To be more precise, assume that we have a polytope P that has parallel faces. For every pair (E; E0) of faces whose affine hulls intersect, choose a (fi- 0 nite) point pEE0 in the intersection of the affine hulls. For every pair (E; E ) of faces whose projective hulls intersect, choose an infinite point qEE0 in the intersection of their projective hulls. Let H be a finite hyperplane having all the points pEE0 on one side, and avoiding all the points qF F 0 . Apply a projective transformation that maps H onto the hyperplane at infinity, to get a new polytope P 0. It is clear that P 0 has no parallel faces, and it is easy to argue that every pair of faces that are antipodal in P 0 correspond to antipodal faces in P . Hence (1) implies the theorem. We now prove (1). Let P be a convex polytope in Rn having no parallel faces.

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